29-limit: Difference between revisions

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'''29-limit''' is the 10th [[prime limit]] and is ~~thus~~ a superset of the [[23-limit]] and a subset of the [[31-limit]]. The prime 29 is notable as being the prime that ends a record prime gap starting at 23. Thus, the 29-limit is in some sense analogous to the [[11-limit]] as both include the prime ending a record prime gap.

The '''29-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 29. It is the 10th [[prime limit]] and is a superset of the [[23-limit]] and a subset of the [[31-limit]]. The prime 29 is notable as being the prime that ends a record prime gap starting at 23. Thus, the 29-limit is in some sense analogous to the [[11-limit]] as both include the prime ending a record prime gap.

~~[[282edo|282EDO]]~~ is the ~~smallest EDO that is consistent~~ to the 29~~-odd~~-limit~~. Intervals~~ [[~~29/16~~]] ~~and [[32/29]] are very accurately approximated by [[7edo|7EDO]] (1\7 for 32/29~~, ~~6\7 for 29/16)~~.

The 29-limit is a rank-10 system, and can be modeled in a 9-dimensional lattice, with the primes 3 to 29 represented by each dimension. The prime 2 does not appear in the typical 29-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a tenth dimension is needed.

== ~~See also~~ ==

These things are contained by the 29-limit, but not the 23-limit:

* [[~~Harmonic~~ limit]]

* The [[29-odd-limit]];

* [[29-odd-limit]]

* Mode 15 of the harmonic or subharmonic series.

The 29-limit intervals of the 2.3.29 subgroup are [[submajor and supraminor]], with [[29/27]] being a supraminor second, [[32/29]] a submajor second, [[29/24]] a supraminor third, and [[36/29]] a submajor third, with their [[octave complement]]s classified accordingly. While supraminor and submajor intervals occur in lower limits, such as [[14/13]], [[11/10]], and [[17/14]], these combine multiple primes higher than 3, unlike the 29-limit ones. The [[29/1|29th harmonic]] is thus quite simple to classify by [[5L 2s|diatonic]] classification, and has a characteristic [[interval quality]] like harmonics [[5/1|5]], [[7/1|7]], etc. Primes [[17/1|17]] and [[23/1|23]] are not so friendly in terms of interval categorization, and may be considered discordant to the fundamental, being a semitone and a tritone when [[octave reduced]] respectively. Thus many people wish to exclude them, leading to the 2.3.5.7.11.13.19.29 subgroup.

However, the 29-limit approaches the point where [[consonance]] stops being registered, and intervals become very close to each other, such as [[29/28]] only being wider than [[30/29]] by [[841/840]], a comma of 2.06{{c}}. This difference is [[JND|unnoticeable]] melodically, and very difficult to hear harmonically.

== Edo approximations ==

[[282edo]] is the smallest edo that is [[consistent]] to the [[29-odd-limit]]. [[1323edo]] is the smallest edo that is [[distinctly consistent]] to the 29-odd-limit. The intervals [[29/16]] and [[32/29]] are very accurately approximated by [[7edo]] (1\7 for 32/29, 6\7 for 29/16).

Edos with increasingly better approximations of the 29-limit ([[monotonicity limit]] ≥ 29 and decreasing [[TE error]]): {{EDOs| 72, 77, 99ef, 118, 121i, 130, 140, 152fgj, 159, 183, 217, 243e, 270, 282, 311, 422, 472, 494h, 525, 535, 540, 554e, 566gj, 571, 581, 581j, 624j, 653, 692i, 718, 742i, 814, 882, 908, 954hj, 1106, 1282, 1308, 1323, 1395, 1578 }}, etc. For a more comprehensive list, see [[Sequence of equal temperaments by error]].

{{Note| [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "99ef" means taking the second closest approximations of harmonics 11 and 13. }}

== Music ==

; [[Francium]]

* [https://www.youtube.com/watch?v=PwvKS0RhTgs ''Spring Your Miracle''] (2026)

; [[Randy Wells]]

* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021)

[[Category:29-limit| ]]

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 {{Prime limit navigation|29}}
-'''29-limit''' is the 10th [[prime limit]] and is thus a superset of the [[23-limit]] and a subset of the [[31-limit]]. The prime 29 is notable as being the prime that ends a record prime gap starting at 23. Thus, the 29-limit is in some sense analogous to the [[11-limit]] as both include the prime ending a record prime gap.
+The '''29-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 29. It is the 10th [[prime limit]] and is a superset of the [[23-limit]] and a subset of the [[31-limit]]. The prime 29 is notable as being the prime that ends a record prime gap starting at 23. Thus, the 29-limit is in some sense analogous to the [[11-limit]] as both include the prime ending a record prime gap.
-[[282edo|282EDO]] is the smallest EDO that is consistent to the 29-odd-limit. Intervals [[29/16]] and [[32/29]] are very accurately approximated by [[7edo|7EDO]] (1\7 for 32/29, 6\7 for 29/16).
+The 29-limit is a rank-10 system, and can be modeled in a 9-dimensional lattice, with the primes 3 to 29 represented by each dimension. The prime 2 does not appear in the typical 29-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a tenth dimension is needed.
-== See also ==
+These things are contained by the 29-limit, but not the 23-limit:
-* [[Harmonic limit]]
+* The [[29-odd-limit]];
-* [[29-odd-limit]]
+* Mode 15 of the harmonic or subharmonic series.
+The 29-limit intervals of the 2.3.29 subgroup are [[submajor and supraminor]], with [[29/27]] being a supraminor second, [[32/29]] a submajor second, [[29/24]] a supraminor third, and [[36/29]] a submajor third, with their [[octave complement]]s classified accordingly. While  supraminor and submajor intervals occur in lower limits, such as [[14/13]], [[11/10]], and [[17/14]], these combine multiple primes higher than 3, unlike the 29-limit ones. The [[29/1|29th harmonic]] is thus quite simple to classify by [[5L 2s|diatonic]] classification, and has a characteristic [[interval quality]] like harmonics [[5/1|5]], [[7/1|7]], etc. Primes [[17/1|17]] and [[23/1|23]] are not so friendly in terms of interval categorization, and may be considered discordant to the fundamental, being a semitone and a tritone when [[octave reduced]] respectively. Thus many people wish to exclude them, leading to the 2.3.5.7.11.13.19.29 subgroup.
+However, the 29-limit approaches the point where [[consonance]] stops being registered, and intervals become very close to each other, such as [[29/28]] only being wider than [[30/29]] by [[841/840]], a comma of 2.06{{c}}. This difference is [[JND|unnoticeable]] melodically, and very difficult to hear harmonically.
+== Edo approximations ==
+[[282edo]] is the smallest edo that is [[consistent]] to the [[29-odd-limit]]. [[1323edo]] is the smallest edo that is [[distinctly consistent]] to the 29-odd-limit. The intervals [[29/16]] and [[32/29]] are very accurately approximated by [[7edo]] (1\7 for 32/29, 6\7 for 29/16).
+Edos with increasingly better approximations of the 29-limit ([[monotonicity limit]] ≥ 29 and decreasing [[TE error]]): {{EDOs| 72, 77, 99ef, 118, 121i, 130, 140, 152fgj, 159, 183, 217, 243e, 270, 282, 311, 422, 472, 494h, 525, 535, 540, 554e, 566gj, 571, 581, 581j, 624j, 653, 692i, 718, 742i, 814, 882, 908, 954hj, 1106, 1282, 1308, 1323, 1395, 1578 }}, etc. For a more comprehensive list, see [[Sequence of equal temperaments by error]].
+{{Note| [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "99ef" means taking the second closest approximations of harmonics 11 and 13. }}
+== Music ==
+; [[Francium]]
+* [https://www.youtube.com/watch?v=PwvKS0RhTgs ''Spring Your Miracle''] (2026)
+; [[Randy Wells]]
+* [https://www.youtube.com/watch?v=4RsACF6s-5U ''Cloud Aliens''] (2021)
 [[Category:29-limit| ]] <!-- main article -->